A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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8 views

Finding FAR and FRR from outlier score

I have a question to raise. Is there anyone here who know who to derive the FAR (False Acceptance Rate) and FRR (False Rejection Rate) using the outlier scores? I am trying to compare the unsupervised ...
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12 views

Stochastic Integration with respect to Cauchy Process?

I'm interested in a one-dimensional stochastic process: $$dX_t = f(X_t)dt + g(X_t) dZ_t$$ where $Z_t$ is a Cauchy process and $f,g$ are nice polynomials (I'm looking at an ODE that gets perturbed by ...
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13 views

Covariance of two geometric Brownian motions different tenors

Assume we have a geometric Brownian motions $dX_t=rX_tdt+σXtdW_1t$ I am want to see the $ Cov(X_T,X_S) $ where S > T by definition: $ E[X_T X_S] = X_0^2 exp(r-σ^2/2)(T+S)) E[exp(σ(W_S+W_T))] $ ...
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1answer
30 views

Jump Diffusion Infinitesimal generator

I have this difussion process $dX(t)=\mu X(t)dt+\sigma X(t)dW(t)+u X(t) dN(t),\qquad X(0)=x > 0$ where $W(t)$ is a Brownian Motion and $N(t)$ is a Poisson process. And I need to know the ...
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1answer
25 views

Is multiplication of a correlated random variable and a independent random variable, an independent random variable

I have a random variable that is a multiplication of two random variables as bellow: $$A_n=B_n\times C_n$$ $B_n$s are identically distributed with zero mean and are correlated for different $n$s and ...
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1answer
22 views

Proving linear operators are Markov Generators

I am trying to do the following question from Liggett. Let $A$ be a linear operator defined on the space of continuous functions $C(E)$ for a compact set $E$. Define $A$ by $A=T-I$ where $T$ is a ...
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26 views

proof of the stratonovich integral?

solving the integral $\int \Phi(x_t,t)dx_t$ writing the equation in the form we can write the integaral as the mean square limit $$\int \Phi(x_t,t)dx_t=\lim_{\Delta \to 0} \sum^{j=1}_{N-1} ...
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1answer
21 views

Lévy Process existence of the expectation of the supremum of the past process.

Given a Lévy Process $X_{t}$ in $\mathbb{R}^{d}$, with $X_{t}^{*}:=\sup_{s\in[0,t]}|X_{s}|$. I want to show, that for $t>0$ with $E[|X_{t}|]<\infty$ for $t>0$, then ...
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19 views

To estimate the probability that a diffusion reaches a certain value

I have a diffusion process define by the following equation: \begin{equation} dX_t=X_t[\beta(N-X_t)-\alpha]dt+\sqrt{X_t(\beta(N-X_t)+\alpha}) { }dB_t \end{equation} and I proved that the solution ...
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42 views

Compute $\frac{d}{dt}\int_0^t e^{x(s)}ds$, where $x$ is a standard Brownian motion.

How to compute the following differentiation? Is there a general rule that can be applied? $$\frac{d}{dt}\int_0^t e^{x(s)}ds$$ in the case of $x=W$ where $W$ is a standard brownian motion, is there ...
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22 views

Are continuous processes almost surely bounded?

Is any process with continuous sample paths almost surely bounded on a finite horizon? If this is true, let $\{X_t\}_{t \in [0, T]}$ be such a process with continuous sample paths. Then we have $|X_t| ...
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26 views

For a stopping time $T$, prove that $X^T_t = \mathbb{E}\left[X_T\mid \mathcal{F}_t\right]$

We have a sigma-algebra $\mathcal{F}=\mathcal{F}_{\infty}$, a stopping time $T$ and an integrable random variable $X$ and define a martingale by $X_t = \mathbb{E}[X \mid \mathcal{F}_t], ...
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0answers
39 views
+50

Simple random walk on $\mathbb Z^d$ and its generator

I'm still trying to figure out definitions and properties of random walks on $\mathbb Z^d$. My goal is to work up to understanding some large deviation principles for the local times of such random ...
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1answer
14 views

Reformulation of SLLN with continuous nondecreasing process as time

Given a filtered probability space $(\Omega,\mathcal{F}_{t},P)$ and a continuous nondecreasing process $U_{t}$ with $U_{0}=0$ and $U_{t}\rightarrow \infty$ $P-a.s.$ as $t$ goes to $\infty$. Given a ...
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31 views
+250

Finding where the tail starts for a probability distribution, from its generating function

Suppose we generate "random strings" over an $m$-letter alphabet, and look for the first occurrence of $k$ consecutive identical digits. I was with some effort able to find that the random variable ...
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1answer
51 views

How to put my knowledge of probability and statistics to practice

Background: I am a masters student in stochastic analysis. My course is very theoretical, which in general is fine by me, it is what I enjoy the most. From the more data-friendly subjects, I have (or ...
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1answer
25 views

Moments of a random sum with bounds Poisson distributed?

We have that $N$ and ${X_1,X_2,\dots}$ are all independent and that $f(x)=Cx^2(1-x)^2$. Then, we have: $$Z=\sum_{j=1}^{N+1}X_j$$ $N$~Poisson$\lambda$. Find the expectation and the variance of $Z$. ...
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0answers
71 views

Kolmogorov continuity theorem on Wikipedia.

I am wondering if the Kolmogorov continuity theorem on Wikipedia is wrong? : They say that the modification is sample continuous, and when we click on that link it says that it is a.s. continuous. ...
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1answer
25 views

Example of a non square-integrable martingale?

Are there (simple) examples of martingales which aren't square integrable?
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17 views

What are some modern books on Markov Chains with plenty of good exercises?

I would like to know what books people currently like in Markov Chains (with syllabus comprising discrete MC, stationary distributions, etc.), that contain many good exercises. Some such book on ...
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4answers
68 views

$2$ players take turns and draw from a box containing $1000$ balls, $3$ of them are black.

I'm not sure how to tackle this question. Assume a box containing $1000$ balls, $3$ of them are black and the rest are white. $2$ players $A_1$ & $A_2$ take turns and draw from the box without ...
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1answer
69 views

If $dX_{t} = X_{t}\,dt + \,dB_{t}$, why does $e^{- t}dX_{t} = e^{-t} X_{t} \,dt + e^{-t} \,dB_{t}$?

I'm taking a course in stochastic differential equations, and in order to solve $dX_{t} = X_{t}\,dt + \,dB_{t}$, the book gives a hint: to multiply both sides of this equation by $e^{-t}$. (But, as ...
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1answer
44 views

Which inequalities are there with stochastic integration?

Which inequalities can I use with stochastic integration? For example, with the standard lebesgue integral we have $$\left|\int_\Omega f(x) dx\right| \le M |\Omega|$$ (where $M$ is the maximum of ...
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1answer
39 views

How to show that this is a martingale?

Let $H_s$ be a predictable and bounded process. How can I show that $$M_t = \int_0^t H_s \, dW_s$$ is a martingale? Clearly since $H_s \in L^2_\text{loc} (W)$ we have that $M_t$ is a local ...
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27 views

The “how many pieces do you have buy on average” problem, a markov problem?

I recently discovered a problem similar to this one in a book about Markov chains: Assume you can buy $n-$ different set of cards in a store, but you do not know which one you'll buy: What is the ...
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34 views

Are there different definitions of a continuous time Markov chain, condition on a finite or infinite number of earlier values?

My book defines a continuous time Markov chain like this. Let $\{X_t\}, t \in T$ be a stochastic process on $(\Omega, \mathcal{A}, P)$, with a countable state space $S$. The process is a Markov ...
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1answer
34 views

Poisson Process with continuous rate, Finding Conditional Number of Arrivals

Poisson with customer arrival to the shop rate given by $\lambda (t)=16-(t-4)^2$ Calculate $P(N(5)-N(3)=40|N(4)=70)$ where $N(i)$ means the number of arrivals in the first $i$ hours. The shop ...
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35 views

Markov Chain: Steady State Distribution.

A total of $M$ balls are divided between two urns A and B. A ball is chosen uniformly at random. If it is chosen from urn A then it is placed in urn B with probability $b$ and otherwise it is returned ...
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1answer
19 views

What is the difference between a martingale and doob's martingale?

Every sequence that was termed as a doob's martingale, I was able to deduce that it was also a martingale. So here are few of my questions: 1) Is it correct to say that every doob martingale is also ...
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1answer
22 views

Prove a conditional distribution is uniformly distributed across a given interval?

$X$ and $Y$ are independent random variables identically exponentially distributed with $\lambda$. Take $Z=X+Y$. Show that $(X|Z=z)$ is uniformly distributed over $(0<x<z)$. Then, find ...
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1answer
18 views

Doubt with Notation on Conditional Expected Value Demonstration

I´m having trouble writing a demonstration for the Conditional Expected Value using $\sigma$-algebra. I know its really simple and actually logic but I just can´t find the way to write it. Hope anyone ...
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17 views

Continuity of random variable as function of a random variable

Suppose, we are given a continuos random variable $X$ and a continuous and nondecreasing function $f$. Can it be shown that a second random variable $Y=f(X)$ is continuos on the support of $X$? What ...
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1answer
30 views

Marginal distribution from a Poisson distribution where intensity is exponentially distributed?

Given that $N$ is Poisson distributed with a random intensity $Y$, the conditional distribution of $(X|Y)$ is defined as, for $n=0,1,\dots$ $$P[N=n|Y=\lambda]=e^{-\lambda}\lambda^n\frac{1}{n!}$$ $Y$ ...
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1answer
29 views

Poisson Probability with rate $\lambda (t)=-(t-4)^2+16$

The rate at which customer arrive to the bookstore is $\lambda (t)=-(t-4)^2+16 $ where $t$ measured in hours. The customers can buy a book with probability $0.5$ and they can also buy a coffee ...
2
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1answer
35 views

Probability matrices in an online game or how to approach matching players to maps to achieve better user experience

Probably I had nonstandart question, but I hope to find some help and valueable advice. Assume I have an online game with $n$ players (let's say $n$ is about 100.000). There's also $m$ maps ($m$ is ...
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1answer
20 views

Random sum of normal distribution with bounds Poisson distributed?

A random variable, $M$, is Poisson distributed with $\lambda=2$. ${X_1,X_2,\dots}$ are independently identically distributed random variables with $\mu=3$ and $\sigma=.2$. Introduce a new random ...
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1answer
23 views

Stochastic process independent of its future

Are there examples of predictable stochastic processes $X$ such that their past is independent of their future? More formally, such that $\sigma\{X_s | s\in (0,t]\}$ is independent of $\sigma\{X_s | ...
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0answers
20 views

Ergodicity of the natural measure implies uniqueness of the invariant density?

Consider a dynamical system $x_{t+1} = F(x_t)$ defined in $\Omega$ and its natural measure to be $\mu$. The Perron-Frobenius operator $F$ maps the density $f$ in time according to $f_{t+1} = F(f_t)$. ...
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0answers
28 views

Ito Formula for Stochastic Integral

Suppose I have $$dS_t = \mu(S_t,t) dt + \sigma(S_t,t)dW_t$$ What would be the process satisfying the following process of $y_t$? $$y_t = \int_0^t S_u du + \int_0^t S_u dW_u$$ I'm not quite sure ...
2
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1answer
13 views

How to prove that the following process is a Martingale using Ito's formula?

I am asked to prove that $Y_t$ is a martingale where $Y_t=\exp\left(\int_0^tf(s)\,dW_s-1/2\int_0^tf(s)^2\,dt\right)$ using Ito's formula. After applying Ito's formula (I hope I made no mistake) I get ...
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2answers
26 views

If $M_t$ is a martingale, is this process a martingale too?

If $M_t$ is a martingale, is this process a martingale too ? $X_t=\int_0^tY_sdM_s$ where $Y_t$ is some process that makes $\int_0^tY_sdM_s$ defined If not, what about the case $Y_t=M_t$ ?
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28 views

Explicit formula for return probability of simple random walk

Is there an explicit formula for the probability that a simple symmetric random walk on $\mathbb{Z}$ starting from $1$ will not hit $0$ before time $t$?
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23 views

Return time for two independent one dimensional random walks

Let $X^1$ and $X^{-1}$ be two simple random walk in $\mathbb{Z}$ starting respectively from $1$ and $-1$. Let $\tau$ be the first time one of them reaches the origin, $$\tau = \inf \{ j \geq 0 \, : ...
2
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0answers
56 views

Proof of the existence of a reversible stationary distribution

$p$ is a finite Markov chain where $p(i,j)>0$ for all $i,j$. Prove a reversible stationary distribution exists for $p$ if $p(i,j)p(j,k)p(k,i)=p(i,k)p(k,j)p(j,i)$ for all $i,j,k$ This question is ...
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15 views

Convergence of stochastic processes via convergence of infinitesimal generators

Given a sequence of sequence processes $(X_N(\cdot))_{N \geq 0}$, I want to show this sequence converges to another process $X(\cdot)$ by considering that the sequence of generators $(A_N)_{N \geq 0}$ ...
4
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1answer
44 views

A counterexample for supremum of stopping times

Let $\mathbb{F} = \{ \mathcal{F}_t \}_{t \geq 0}$ be a continuous time filteration. $\tau : \Omega \to [0, \infty]$ is called an $\mathbb{F}$-stopping time if $\{ \tau \leq t \} \in \mathcal{F}_t$ for ...
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23 views

Distribution of Hitting Times

I am curious about the following problem: Let the diffusion process $\{X_t\}_{t\ge 0}$ be defined as $$dX_t=c(1-X_t)X_td B_t$$ where $X_0\in (a,b)\subseteq (0,1)$, $c>0$, and $B_t$ is the standard ...
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0answers
16 views

Is the result of a Monte-Carlo simulation of a continuous function and with continuous input distributions again continuous?

Is the result of a Monte-Carlo simulation of a continuos function and with continuos input distributions again continuous? Suppose, we have a continuos function $f$ and a number of continuous random ...
2
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2answers
59 views

Modeling with Markov Chains and one-step analysis

I have set up the following model: Let $X_n$ be the number of heads in the $n$-th toss and $P(X_0=0)=1$. I can calculate the transition matrix $P$. Define $$ T=\min\{n\geq 0\mid X_n=5\}. $$ Then ...
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0answers
33 views

Dealing with Recurrence Relations of Random Variables

Let $(Y_n)_{n\in \mathbb N} $ be some sequence of independent random variables, and $(X_n)_{n\in \mathbb N} $ another sequence, defined recursively as follows: $$X_{n+1} = \alpha X_n + \beta Y_n ...